Parameterized Complexity of s-Club Cluster Edge Deletion: When Is the Diameter Bound Necessary?

We study when the diameter bound s is essential for the tractability of the s-Club Cluster Edge Deletion problem. Given a graph G = (V, E) and integers k and s, the goal is to delete at most k edges so that every connected component of the resulting graph has diameter at most s. This problem generalizes Cluster Edge Deletion (s = 1) and captures distance-bounded clustering tasks. Montecchiani et al. (Information and Computation, 2023) proved that the problem is fixed-parameter tractable when parameterized by s + tw(G) and asked whether dependence on s is necessary. We answer negatively by showing W[1]-hardness when parameterized by pathwidth (and hence by treewidth), proving that s cannot in general be dropped. On the positive side, we show FPT algorithms parameterized by treedepth, neighborhood diversity, or cluster vertex deletion number, extending results of Italiano et al. (Algorithmica, 2023) and Komusiewicz and Uhlmann (SOFSEM, 2011). We also show that no polynomial kernel exists when parameterized by vertex cover number, even for s = 2. Classically, the problem is NP-hard on split graphs for s = 2, complementing the polynomial case s = 1. We give an FPT bicriteria approximation scheme running in f(k, 1/epsilon) * n^{O(1)} that outputs a set of at most k deletions whose components have diameter at most (1 + epsilon)s. Finally, we introduce a directed generalization, s-Club Cluster Arc Deletion, extending the undirected case to reachability distances, and show it is W[1]-hard in parameter k even on directed acyclic graphs (DAGs).